NameMath 221 First Midterm Exam Thursday October 5 2006Circle your section:301 7:45 MW B305 VAN VLECK Murcko302 8:50 MW B333 VAN VLECK Ganguly303 8:50 MW B329 VAN VLECK Andrejko304 9:55 MW B219 VAN VLECK Andrejko305 9:55 MW B105 VAN VLECK Hu306 11:00 MW B105 VAN VLECK Hu308 12:05p MW B329 VAN VLECK Ganguly310 2:25p MW B203 VAN VLECK Murcko312 9:55-11:50 MWF 277 BASCOM Rouse313 12:15-2:10 MWF 3349 ENGR HALL OwenI 40 PointsII 30 PointsIII 30 PointsIV 30 PointsV 25 PointsVI 25 PointsVII 20 PointsTotal 200 PointsShow your reasoning.I. (40 points.) (a) Findd2ydx2if y = 5x.(b) Finddwdxif w = xx.(c) Finddudxif u = sin(ln x).(d) Find limx→∞ex− 1ex+ 1.II. (30 points.) Consider the parametric curvex =1 − t21 + t2, y =2t1 + t2.(i) Show that each point on this curve lies on the unit circle.(ii) What point corresponds to the parameter value t = 2?(iii) Write the equation of the tangent line at this point in parametric form:x = x0+ a(t − t0), y = y0+ b(t − t0).(iv) Write the equation of the tangent line at this point in point slope form:y = y0+ m(x − x0).III. (30 points.) You are videotaping a race from a stand 120 feet from the track, followinga car that is moving at a constant velocity along a straight track. When the car is directly infront of you the camera angle is changing at a rate of π/3 radians per second. How fast is thecar going? How fast will the camera angle be changing a half second later?IV. (30 points.) Denote the linear approximations to f near a and to g near b byL(x) = f(a) + f0(a)(x − a), M(y) = g(b) + g0(b)(y − b),and assume b = f(a). Show thatM ◦ L(x) = (g ◦ f)(a) + (g ◦ f)0(a)(x − a).(This says that The linear approximation to the composition is the composition of the linearapproximations.)V. (25 points.) Water runs out of a cylindrical tank from a drain in the bottom. The waterlevel in the tank t hours after the tank starts to drain isy = 61 −t122metersand the tank drains completely after 12 hours. Show that the rate at which the water level dropsis proportional to the square root of the water level and find the constant of proportionality.VI. (25 points.) (i) Write the polynomial of degree four which best approximates exnear x = 0.(i) What is the precise definition of the phrase “the polynomial of degree four which bestapproximates exnear x = 0”.(i) Write the polynomial of degree eight which best approximates ex2near x = 0.VII. (20 points.) (a) The functionf(x) =x + c for x < 13xfor x ≥ 1is continuous. What is c?(b) Draw a crude graph of the equation y = f (x) where f(x) is as in part (a).(c) Give a formula (like the above formula for f(x)) for the inverse function x = f−1(y) of thefunction y = f(x) of part
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